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Numbers n such that n^2 + 1 is not divisible by k^2 + 1 for any k in [1,n-1].
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%I #17 Sep 18 2019 09:52:29

%S 1,2,4,6,10,14,16,20,24,26,34,36,40,44,46,50,54,56,60,66,70,74,76,84,

%T 86,90,94,96,100,104,110,114,116,120,124,126,130,134,136,144,146,150,

%U 156,160,164,170,176,180,184,186,190,194,196,204,206,210,214,220,224

%N Numbers n such that n^2 + 1 is not divisible by k^2 + 1 for any k in [1,n-1].

%C Equivalently, A066743(n)=1.

%C If n^2 + 1 is prime, n is in the sequence; i.e., the sequence contains A005574. But so are many other values of n: 34,44,46,50,60,70,76,86,96,...

%H Harry J. Smith, <a href="/A066755/b066755.txt">Table of n, a(n) for n = 1..1000</a>

%p a:= proc(n) option remember; local k; for k from 1+

%p `if`(n=1, 0, a(n-1)) while ormap(t->

%p irem(k^2+1, t)=0, [(j^2+1)$j=1..k-1]) do od; k

%p end:

%p seq(a(n), n=1..100); # _Alois P. Heinz_, Sep 18 2019

%t a66743[ n_ ] := Length[ Select[ Range[ 1, n ], IntegerQ[ (n^2+1)/(#^2+1) ]& ] ]; Select[ Range[ 1, 300 ], a66743[ # ]==1& ]

%o (PARI) { n=0; for (m=1, 10^10, k=1; b=1; t=m^2 + 1; while (k < m - 1, if (t%(k^2 + 1)==0, b=0; break); k++); if (b, write("b066755.txt", n++, " ", m); if (n==1000, return)) ) } \\ _Harry J. Smith_, Mar 23 2010

%Y Cf. A066743, A005574.

%K nonn

%O 1,2

%A _Benoit Cloitre_, Jan 16 2002

%E Edited by _Dean Hickerson_, Jan 20 2002